So if I'm renting a dedicated server just like I'd be renting a taxi, am I also "serverless"?
In both cases there is a lot of grey areas, but the general concept is the same: If you don't own, maintain or operate (drive) a server/car, you can call yourself server/car-less, regardless of whether a server or car is still being used.
You're renting it long term? That's like leasing a very specific car, so no not really.